Null Curves in $\mathbb{C}^3$ and Calabi-Yau Conjectures

For any open orientable surface $M$ and convex domain $\Omega\subset \mathbb{C}^3,$ there exists a Riemann surface $N$ homeomorphic to $M$ and a complete proper null curve $F:N\to\Omega.$ This result follows from a general existence theorem with many applications. Among them, the followings: For any convex domain $\Omega$ in $\mathbb{C}^2$ there exist a Riemann surface $N$ homeomorphic to $M$ and a complete proper holomorphic immersion $F:N\to\Omega.$ Furthermore, if $D \subset \mathbb{R}^2$ is a convex domain and $\Omega$ is the solid right cylinder $\{x \in \mathbb{C}^2 | {Re}(x) \in D\},$ then $F$ can be chosen so that ${\rm Re}(F):N\to D$ is proper. There exists a Riemann surface $N$ homeomorphic to $M$ and a complete bounded holomorphic null immersion $F:N \to {\rm SL}(2,\mathbb{C}).$ There exists a complete bounded CMC-1 immersion $X:M \to \mathbb{H}^3.$ For any convex domain $\Omega \subset \mathbb{R}^3$ there exists a complete proper minimal immersion $(X_j)_{j=1,2,3}:M \to \Omega$ with vanishing flux. Furthermore, if $D \subset \mathbb{R}^2$ is a convex domain and $\Omega=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 | (x_1,x_2) \in D\},$ then $X$ can be chosen so that $(X_1,X_2):M\to D$ is proper. Any of the above surfaces can be chosen with hyperbolic conformal structure.